Proving Lines Parallel Perpendiculars and Distance
Converse of Corresponding Angles If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
Parallel Postulate If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.
Theorems
Examples Determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. ∠1 ≅ ∠6
Examples Determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. ∠2 ≅ ∠3
Examples Find m∠MRQ so that a ∥ b.
Examples Find m∠MRQ so that a ∥ b. Alt. Int. Angles 5x + 7 = 7x – 21
Perpendiculars and Distance The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point.
Perpendicular Postulate If given a line and a point not on the line, then there exists exactly one line through the point that is perpendicular to the given line.
Examples Line l contains points at (-5, 3) and (4, -6). Find the distance between line l and point P(2, 4).
Examples Line l contains points at (-5, 3) and (4, -6). Find the distance between line l and point P(2, 4). x: 4 units y: 4 units distance is √(42 + 42) = √32 = 4 √2
Examples Line l contains points at (1, 2) and (5, 4). Construct a line perpendicular to l through P(1, 7). Then find the distance from P to l.
Examples Line l contains points at (1, 2) and (5, 4). Construct a line perpendicular to l through P(1, 7). Then find the distance from P to l. x: 2 units y: -4 units distance is √(22 + 42) = √20 = 2 √5
Theorem In a plane, if two lines are each equidistant from a third line, then the two lines are parallel to each other.
Examples Find the distance between the parallel lines r and s whose equations are y = -3x – 5 and y = -3x + 6, respectively.
Examples Find the distance between the parallel lines r and s whose equations are y = -3x – 5 and y = -3x + 6, respectively. m⊥= 1/3 point = (-2, 1)